Conclusions and recommendations

1. A complete range of shell shapes and loading has been studied. The shells buckle in ring mode (1-3, 3-3, 2-1, 2-3), column mode (2-2), mixed column-ring mode (1-1) and inextensional mode (3-2, 3-1, 1-2)(page 19).

2. The buckling formula ncr = -0.6 Et^2/a is accurate for ring, column and mixed mode buckling. This can be used to identify inextensional buckling: for inextensional buckling a linear buckling analysis gives a much smaller load than the formula (page 22).

3. The conclusions of this report are valid for large and small shells. This follows from the fact that the linear buckling load factors do not change when all dimensions are halved, while Young's modulus and the surface load are doubled (page 38).

4. The transition in buckling shape from column buckling to ring buckling occurs gradually for increased radial tension. However, the transition in linear buckling load factor is abrupt (page 42).

5. The inward and outward shape imperfections give same buckling load factor (page 52). They would be different, if the buckling length were large compared to the shell length or radius.

6. Two of the inextensional modes become extensional in the non-linear analysis (page 52). The buckling formula does not predict n_cr but buckling formula combined with knockdown factor does predict the n_ult. Only truely inextensional is negative Gaussian curvature with double compression (3-2).

7. A cylinder with just axial load is an exception. It has a smaller knockdown factor than the other shapes and loading (page 52).

8. The knockdown factor should not be applied to truely inextensional buckling mode (page 62).

9. The knockdown factor formula sometimes has three solutions, of which one is physically realistic (page 83).

10. The knockdown factor formula produces reasonable values. However, the formula is not accurate (page 62).

11. The knockdown factor does not depend on the curvature ratio kyy/kxx or the membrane force ratio nxx/nyy. It only depends on the imperfection amplitude and d/t (page 65) and the slenderness a/t (page 6). 

12. In several nonlinear analyses, divergence occurred already at load step 6. Therefore, the maximum error of the nonlinear buckling load factors is 0.5/5.5 = 9% (page 51). It is recommended to improve this in continuing research.